Abstract
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem that models a turbulent flow in a porous medium. The obtained inequalities are used to obtain a lower bound for the eigenvalues of corresponding equations.
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1 Introduction
The p-Laplacian operator arises in different mathematical models that describe physical and natural phenomena (see, for example, [1–6]). In particular, it is used in some models related to turbulent flows (see, for example, [7–9]).
In this paper, we present some Lyapunov-type inequalities for a fractional-order model for turbulent flow in a porous medium. More precisely, we are interested with the nonlinear fractional boundary value problem
where \(2<\alpha\leq3\), \(1<\beta\leq2\), \(D_{a^{+}}^{\alpha}\), \(D_{a^{+}}^{\beta}\) are the Riemann-Liouville fractional derivatives of orders α, β, \(\Phi_{p}(s)=|s|^{p-2}s\), \(p>1\), and \(\chi: [a,b]\to\mathbb{R}\) is a continuous function. Under certain assumptions imposed on the function q, we obtain necessary conditions for the existence of nontrivial solutions to (1.1). Some applications to eigenvalue problems are also presented.
For completeness, let us recall the standard Lyapunov inequality [10], which states that if u is a nontrivial solution of the problem
where \(a< b\) are two consecutive zeros of u, and \(\chi: [a,b]\to \mathbb{R}\) is a continuous function, then
Note that in order to obtain this inequality, it is supposed that a and b are two consecutive zeros of u. In our case, as it will be observed in the proof of our main result, we assume just that u is a nontrivial solution to (1.1).
Inequality (1.2) is useful in various applications, including oscillation theory, stability criteria for periodic differential equations, and estimates for intervals of disconjugacy.
Several generalizations and extensions of inequality (1.2) to different boundary value problems exist in the literature. As examples, we refer to [11–16] and the references therein.
Recently, some Lyapunov-type inequalities for fractional boundary value problems have been obtained. Ferreira [17] established a fractional version of inequality (1.2) for a fractional boundary value problem involving the Riemann-Liouville fractional derivative of order \(1<\alpha\leq2\). More precisely, Ferreira [17] studied the fractional boundary value problem
where \(D_{a^{+}}^{\alpha}\) is the Riemann-Liouville fractional derivative of order \(1<\alpha\leq2\), and \(\chi: [a,b]\to\mathbb{R}\) is a continuous function. In this case, it was proved that if (1.3) has a nontrivial solution, then
where Γ is the Euler gamma function. Observe that if we take \(\alpha=2\) in the last inequality, we obtain the standard Lyapunov inequality (1.2).
Ferreira [18] established a fractional version of inequality (1.2) for a fractional boundary value problem involving the Caputo fractional derivative of order \(1<\alpha\leq2\). In both papers [17, 18], the author presented nice applications to obtain intervals where certain Mittag-Leffler functions have no real zeros.
Jleli and Samet [19] studied a fractional differential equation involving the Caputo fractional derivative under mixed boundary conditions. More precisely, they considered the fractional differential equation
under the mixed boundary conditions
or
where \({}^{\mathrm{C}}D_{a^{+}}^{\alpha}\) is the Caputo fractional derivative of order \(1<\alpha\leq2\). For the boundary conditions (1.5) and (1.6), the following two Lyapunov-type inequalities were derived respectively:
and
The same equation (1.4) was considered by Rong and Bai [20] with the fractional boundary condition
where \(0<\beta\leq1\).
For other related results, we refer to [21–23] and the references therein.
The paper is organized as follows. In Section 2, we recall some basic concepts on fractional calculus and establish some preliminary results that will be used in Section 3, where we state and prove our main result. In Section 4, we present some applications of the obtained Lyapunov-type inequalities to eigenvalue problems.
2 Preliminaries
For the convenience of the reader, we recall some basic concepts on fractional calculus to make easy the analysis of (1.1). For more details, we refer to [24].
Let \(C[a,b]\) be the set of real-valued and continuous functions in \([a,b]\). Let \(f\in C[a,b]\). Let \(\alpha\geq0\). The Riemann-Liouville fractional integral of order α of f is defined by \(I_{a}^{0} f\equiv f\) and
where Γ is the gamma function.
The Riemann-Liouville fractional derivative of order \(\alpha>0\) of f is defined by
where \(n=[\alpha]+1\).
Lemma 2.1
(see [24])
Let \(\alpha>0\). If \(D_{a^{+}}^{\alpha}u\in C[a,b]\), then
where \(n=[\alpha]+1\).
Now, in order to obtain an integral formulation of (1.1), we need the following results.
Lemma 2.2
Let \(2<\alpha\leq3\) and \(y\in C[a,b]\). Then the problem
has a unique solution
where
Proof
From Lemma 2.1 we have
for some real constants \(c_{i}\), \(i=1,2,3\), that is,
The condition \(u(a)=0\) yields \(c_{3}=0\). Therefore,
The condition \(u'(a)=0\) implies that \(c_{2}=0\). Then
Since \(u'(b)=0\), we get
Thus,
For the uniqueness, suppose that \(u_{1}\) and \(u_{2}\) are two solutions of the considered problem. Define \(u=u_{1}-u_{2}\). By linearity, u solves the boundary value problem
which has as a unique solution \(u=0\). Therefore, \(u_{1}=u_{2}\), and the uniqueness follows. □
Lemma 2.3
Let \(y\in C[a,b]\), \(2<\alpha\leq3\), \(1<\beta\leq2\), \(p>1\), and \(\frac{1}{p}+ \frac{1}{q}=1\). Then the problem
has a unique solution
where
Proof
From Lemma 2.1 we have
where \(c_{i}\), \(i=1,2\), are real constants. The condition \(D_{a^{+}}^{\alpha}u(a)=0\) implies that \(\Phi_{p}(D_{a^{+}}^{\alpha}u)(a)=0\), which yields \(c_{2}=0\). The condition \(D_{a^{+}}^{\alpha}u(b)=0\) implies that \(\Phi _{p}(D_{a^{+}}^{\alpha}u)(b)=0\), which yields
Therefore,
that is,
Then we have
Setting
we obtain
Finally, applying Lemma 2.2, we obtain the desired result. □
The following estimates will be useful later.
Lemma 2.4
We have
Proof
Differentiating with respect to t, we obtain
Set
and
Clearly,
On the other hand, using the inequality
and the fact that \(\alpha>2\), we obtain
which yields
As consequence, we have
Then \(G(\cdot,s)\) is a nondecreasing function for all \(s\in[a,b]\), which yields
The proof is complete. □
Lemma 2.5
We have
Proof
Observe that \(H(t,s)=G_{t}(t,s)\) for \(\alpha =\beta+1\). Then, from the proof of Lemma 2.4 we have
On the other hand, for all \(s\in[a,b]\), we have
For \(a\leq t\leq s\leq b\), we have
For \(a\leq s< t\leq b\), we have
Let \(s\in[a,b)\) be fixed. Define the function \(\psi:(s,b]\to\mathbb {R}\) by
We have
Using the inequalities
we get
Thus, for all \(t\in(s,b]\), we have
that is,
The proof is complete. □
Now, we are ready to state and prove our main result.
3 Main result
Our main result is the following Lyapunov-type inequality.
Theorem 3.1
Suppose that \(2<\alpha\leq3\), \(1<\beta\leq2\), \(p>1\), and \(\chi: [a,b]\to\mathbb{R}\) is a continuous function. If (1.1) has a nontrivial continuous solution, then
Proof
We endow the set \(C[a,b]\) with the Chebyshev norm \(\|\cdot\|_{\infty}\) given by
Suppose that \(u\in C[a,b]\) is a nontrivial solution of (1.1). From Lemma 2.3 we have
Let \(t\in[a,b]\) be fixed. We have
where
Using Lemma 2.4 and Lemma 2.5, we obtain
Since the last inequality holds for every \(t\in[a,b]\), we obtain
which yields the desired result. □
Corollary 3.2
Suppose that \(2<\alpha\leq3\), \(1<\beta\leq2\), \(p>1\), and \(\chi: [a,b]\to\mathbb{R}\) is a continuous function. If (1.1) has a nontrivial continuous solution, then
Proof
Let
Observe that the function ψ has a maximum at the point \(s^{*}=\frac {a+b}{2}\), that is,
The desired result follows immediately from the last equality and inequality (3.1). □
For \(p=2\), problem (1.1) becomes
where \(2<\alpha\leq3\), \(1<\beta\leq2\), and \(\chi: [a,b]\to\mathbb {R}\) is a continuous function. In this case, taking \(p=2\) in Theorem 3.1, we obtain the following result.
Corollary 3.3
Suppose that \(2<\alpha\leq3\), \(1<\beta\leq2\), and \(\chi: [a,b]\to \mathbb{R}\) is a continuous function. If (3.3) has a nontrivial continuous solution, then
Taking \(p=2\) in Corollary 3.2, we obtain the following result.
Corollary 3.4
Suppose that \(2<\alpha\leq3\), \(1<\beta\leq2\), and \(\chi: [a,b]\to \mathbb{R}\) is a continuous function. If (3.3) has a nontrivial continuous solution, then
4 Applications to eigenvalue problems
In this section, we present some applications of the obtained results to eigenvalue problems.
Corollary 4.1
Let λ be an eigenvalue of the problem
where \(2<\alpha\leq3\), \(1<\beta\leq2\), and \(p>1\). Then
Proof
Let λ be an eigenvalue of (4.1). Then there exists a nontrivial solution \(u=u_{\lambda}\) to (4.1). Using Theorem 3.1 with \((a,b)=(0,1)\) and \(\chi(s)=\lambda\), we obtain
Observe that
and
where B is the beta function defined by
Using the identity
we get the desired result. □
Corollary 4.2
Let λ be an eigenvalue of the problem
where \(2<\alpha\leq3\) and \(1<\beta\leq2\). Then
Proof
It follows from inequality (4.2) by taking \(p=2\). □
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Acknowledgements
The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Al Arifi, N., Altun, I., Jleli, M. et al. Lyapunov-type inequalities for a fractional p-Laplacian equation. J Inequal Appl 2016, 189 (2016). https://doi.org/10.1186/s13660-016-1132-y
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DOI: https://doi.org/10.1186/s13660-016-1132-y